Any answer on MSE is a distraction of me from my work.

My appreciated answers

1. An entropy inequality
   

2. A law of large numbers for bounded processes
   

3. If $X$ is Gaussian, prove that $X-\lfloor X \rfloor \sim U(0,1)$ as its variance becomes large

4. Show that $f(x)$ is a constant function if $\lim\limits_{h\to 0}\frac{1}{h^3}\int_{-h}^{h}f(x+t)\cdot t\,dt=0$ for all $x$
   

5. About the inequality $x^{x^{x^{x^{x^x}}}} \ge \frac12 x^2 + \frac12$
   

6. Prove that two random variates must be negatively correlated
   

7. Why do the triangular numbers initially form long cycles mod $2^k$?
   

8. Is $\sqrt{n}(\bar X_n - \bar X_m)$ tight?

9. $\underset{x\to 1}{\text{lim}}\int_0^x \frac{\sqrt{t} f(t)}{\sqrt{f(x)-f(t)}} \, \mathrm dt=\frac{ \pi }{\sqrt{2}}$
   

10. Given the sequence $a_1=1$,$a_{n+1}=1+\frac{n}{a_n}$, does the sequence $a_n+n-a_n^2$ converges?

11. Measurability and convergence of sequence of functions
    

12. Prove $\lim\limits_{n \to \infty} na_n=0.$
    

13. Convergence in probability for each $P_k$ implies convergence in probability in $\sum_k \lambda_k P_k$
    

14. Generalisation of Dominated Convergence Theorem
    

15. Is this Stochastic Process bounded from above?
    

16. Proof of Hopf Lax formula
    

17. What is the asymptotic expansion of $x_n$ where $x_{n+1} = x_n+1/x_n$?